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efficient satisfies a1"2 - 2x -j- 10 = 0, the second, x- - 4x
 421 = 0, the third, a?  90 = 0, of the respective degrees
Lnagine (if you can) the set of all algebraic numbers. Among
these will be all the positive rational integers 1,2, 3, ... , since
any one of them, say n, satisfies an algebraic equation, x  72 =
0, in which the coefficients (1, and  n) are rational integers.
But in addition 10 these the set of all algebraic numbers will
include all roots of all quadratic equations with rational integer
coefficients, and all roots of all cubic equations with rational
integer coefficients, and so on, indefinitely. Is it not intuitively
evident that the set of all algebraic numbers will contain infi-
nitely more members than its sw&-set of the rational integers
1. 2, 3, ... ? It might indeed be so, but it happens to be false.
Cantor proved that the set of all rational integers 1, 2, 3, ...
contains precisely as many members as the Infinitely more
inclusive' set of all algebraic numbers.
A proof of this paradoxical statement cannot be given here,
but the kind of device - that of 'one-to-one correspondence* -
upon which the proof is based can easily be made intelligible.
This should induce in the philosophical mind an understanding
of what a cardinal number is. Before describing this simple but
somewhat elusive concept it will be helpful to glance at an
expression of opinion on this and other definitions of Cantor's
theory which emphasizes a distinction between the attitudes of
some mathematicians and many philosophers toward all
questions regarding 'number* or 'magnitude".
;A mathematician never defines magnitudes in themselves,
as a philosopher would be tempted to do; he defines their
equality, their sum, and their product, and these definitions
determine, or rather constitute, all the mathematical properties
of magnitudes. In a yet more abstract and more formal manner
he lays dawn symbols and at the same time prescribes the rules
according to which they must be combined; these rules suffice
to characterize these symbols and to give them a mathematical
value. Briefly, he creates mathematical entities by means of
arbitrary conventions, in the same way that the several chess*
men are defined by the conventions which govern their moves